The Contributions of Hilbert and Dehn to Non-Archimedean Geometries and Their Impact on the Italian School (2007)
In this paper we investigate the contribution of Dehn to the development of non-Archimedean geometries. We will see that it is possible to construct some models of non-Archimedean geometries in order to prove the independence of the continuity axiom and we will study the interrelations between Archimedes' axiom and Legendre's theorems. Some of these interrelations were also studied by Bonola, who was one of the very few Italian scholars to appreciate Dehn's work. We will see that, if Archimedes' axiom does not hold, the hypothesis on the existence and the number of parallel lines through a point is not related to the hypothesis on the sum of the inner angles of a triangle. Hilbert himself returned to this problem giving a very interesting model of a non-Archimedean geometry in which there are infinitely many lines parallel to a fixed line through a point while the sum of the inner angles of a triangle is equal to two right angles. Keywords : David Hilbert, Max Dehn, Federico Enriques, Roberto Bonola, Non-Archimedean geometry
...MoreSimilar Citations
Chapter
Marco Toscano;
(2009)
Geometria e fisica: il significato storico-filosofico della relatività di Einstein nel pensiero di Federigo Enriques
(/isis/citation/CBB324997305/)
Article
Argante Ciocci;
(2018)
I manoscritti urbinati di Federico Commandino: una ricognizione delle buste 120 e 121 della Biblioteca Universitaria di Urbino
(/isis/citation/CBB357150047/)
Article
Argante Ciocci;
(2015)
Luca Pacioli e l'Archimede latino
(/isis/citation/CBB290781922/)
Book
Enrico Giannetto;
Giulia Giannini;
(2009)
Da Archimede a Majorana: la fisica nel suo divenire
(/isis/citation/CBB945473809/)
Article
Stillwell, John;
(2014)
Ideal Elements in Hilbert's Geometry
(/isis/citation/CBB001213909/)
Article
Sieg, Wilfried;
(2014)
The Ways of Hilbert's Axiomatics: Structural and Formal
(/isis/citation/CBB001213914/)
Chapter
Corry, Leo;
(2006)
Axiomatics, Empiricism, and Anschauung in Hilbert's Conception of Geometry: Between Arithmetic and General Relativity
(/isis/citation/CBB000800120/)
Chapter
Arcangelo Rossi;
(2017)
Federigo Enriques between popularization and scientific criticism
(/isis/citation/CBB951087422/)
Book
Caterina Genna;
(2021)
Federigo Enriques matematico e filosofo
(/isis/citation/CBB703132259/)
Book
Enriques, Federigo;
Simili, Raffaella;
(2000)
Per la scienza: Scritti editi e inediti
(/isis/citation/CBB000400942/)
Article
Nastasi, Tina;
(2008)
La storia del pensiero scientifico e il suo significato nell'opera di Federigo Enriques
(/isis/citation/CBB001024085/)
Article
Nurzia, Laura;
(1979)
Relazioni tra le concezioni geometriche di Federigo Enriques e la matematica intuizionista tedesca
(/isis/citation/CBB000000080/)
Book
Alberto Cogliati;
(2024)
La geometria non euclidea. Una breve storia dall’antichità a Poincaré
(/isis/citation/CBB276091257/)
Chapter
Giulia Giannini;
(2009)
Poincaré, la nozione di gruppo e il Programma di Erlangen di F. Klein
(/isis/citation/CBB227666735/)
Article
Krol, Z.;
(2006)
Ancient Geometry and Plato's Philosophy on the Base of Pappus' “Comment on the Xth Book of Elements of Euclid”
(/isis/citation/CBB000931646/)
Article
Shoji, Kota;
(2006)
The Contact of Differential Geometry and Continuous Groups of Transformations: W. Killing's “Ueber die Grundlagen der Geometrie”
(/isis/citation/CBB000772578/)
Chapter
Villaggio, Piero;
(2006)
On Enriques's Foundations of Mechanics
(/isis/citation/CBB000774505/)
Chapter
Reich, Karin;
(2008)
Der Desarguessche und der Pascalsche Satz: Hessenbergs Beitrag zu Hilberts Grundlagen der Geometrie
(/isis/citation/CBB000950185/)
Article
Pejlare, Johanna;
(2007)
Torsten Brodén's Work on the Foundations of Euclidean Geometry
(/isis/citation/CBB000773211/)
Book
Gabriele Lolli;
(2016)
Tavoli, sedie, boccali di birra: David Hilbert e la matematica del Novecento
(/isis/citation/CBB100106060/)
Be the first to comment!